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Permanent Magnet Synchronous Machine Model forReal- Time Simulation

A. B. Dehkordi, Student Member, IEEE, A. M. Gole, Senior Member, IEEE, T. L. Maguire, Senior Member, IEEE

Abstract--This paper presents the implementation of a model

for a permanent magnet motor on a real time digital simulator.

The conventional interfaced model approach popular in some

emt-type programs is shown to be inappropriate for real-time

simulators, because it is more susceptible to numerical instability.

The paper introduces an embedded phase domain model which

reduces numerical instabilities.

Both the conventional as well as the embedded phase domain

model models are shown to provide identical results in the steady

state and transient situations; however the conventional model

becomes unstable quickly if the time-step is increased.

This paper also presents the application of this new model to

simulate the machine using the evaluated inductances that have

been evaluated by (FEM) method based modeling.

Keywords: Permanent magnet machines, real time digital

simulator, Electromagnetic transient analysis, embedded model

of the machine, numerical instability.

I. INTRODUCTION

N recent years, Permanent Magnet Synchronous Motors

(PMSMs) are increasing applied in several areas such as

traction, automobiles, robotics and aerospace technology [1].

Accurate digital simulation tools are necessary to evaluate

their field performance particularly when they are driven with

solid-state drives connected to larger electrical networks. One

of the areas of interest is the design of controllers for these

motor drives. In many applications the physical controls have

to be designed and tuned for best performance. If the

simulation of the motor and drive can be implemented in real-

time, it becomes possible to interface the physical

manufacturer-built controller (not its model) and protection

equipment to the simulation using appropriate digital-analog

and analog-digital converters. The real time digital simulator

is a combination of specialized computer hardware and

software designed specifically for the solution of power

system electromagnetically transients in real-time. One such

real-time digital simulator is the RTDS. Since the RTDS

This work has been done with the financial support of RTDS Technologies

Inc.A. B. Dehkordi is with the University of Manitoba, Winnipeg, Canada (e-

mail: [email protected]).

A. M. Gole is with the University of Manitoba, Winnipeg, Canada (e-mail:

[email protected]).

T. L. Maguire is with RTDS Technologies Inc., Winnipeg, Canada (e-mail:

[email protected]).

Presented at the International Conference on Power Systems

Transients (IPST05) in Montreal, Canada on June 19-23, 2005

Paper No. IPST05 - 159

combines the real time operation properties of analogue

simulators with the flexibility and accuracy of digital

simulation programs, there are many areas where this

technology has been successfully applied [2].

Traditionally transient simulation programs use the dq0

based modeling method to model different types of machines

and the machines are interfaced to the network. These

interfaced models have interfacing delays which can cause

numerical stability problems in the presence of other

interfaced components, particularly when the case is running

in real time. Modeling the permanent magnet machines in

RTDS has important industrial applications such as maglev

trains and new generation of automobiles.

This paper presents two different approaches to model the

permanent magnet synchronous machine in RTDS, the

traditional dq0 model of the machine and the embedded phase

domain model. According to second method the machine is

modeled as a set of time-variant mutual inductances. The

machines are compared in the transient and steady state

situations, and in the final step their performance is observed

with an inverter, supplying the AC voltage for the terminals of

the permanent magnet synchronous machines.

II. EQUIVALENT CIRCUIT MODEL OF THE PMSM

A. Structure of the Permanent Magnet Machine

The cross-sectional layout of a surface mounted permanent

magnet motor is shown in Fig.1.

STATOR

MAGNET

ROTOR

CORE

Fig. 1. Structure of the permanent magnet synchronous machine [3]

The stator carries a three-phase winding, which produces a

near sinusoidal distribution of magneto motive force based on

the value of the stator current. The magnets are mounted on

the surface of the motor core. They have the same role as the

I


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field winding in a synchronous machine except their magnetic

field is constant and there is no control on it [3].

B. Equivalent dq0 model of the PM machine

The dq0 equivalent circuit of the PM machine shown in

Fig.2 is similar to the one for the synchronous machine; it has

the armature resistancesR , d and q axis leakage and mutual

inductancessl , mdL and mqL .

0 d

mqLqd

dt

+

+

qV

qi slsR

0 q

mdLdd

dt

+

+

dV

di

mR

mi

fi

fd

dt

+

sR sl

a)

b)

Fig. 2. Dq0 Equivalent circuit model of the PMSM a) D axis b) Q axis

The rotor magnet can be considered as a loop of constant

current source,mi located at the stator direct axis. Any change

in the magnetic flux of the rotor magnet will cause an induced

electromagnetic force, resulting in a circulating current in the

magnet [3].Essentially resistance mR connected across thedirect-axis magnetization inductance

mdL shows this effect

[3]. There is no leakage inductance in the field. The

permeability of the magnet material is almost unity so the air

gap inductance seen by the stator is the same in direct and

quadrature axes and also no saturation will happen inside the

machine.

Fig. 3 shows the demagnetization curve of the magnet that

can be divided into three regions by three lines, called no-

load, rated-load and excessive-load lines [4].

Fig. 3. Demagnetization curve of the permanent magnet [4]

We always try to not enter the excessive load region;

otherwise the magnet is in danger of being damaged.

The equations for the dq0 model of the Permanent Magnet

Synchronous Machine are:

dd s d q

f

m m m f

q

q s q d

dV R i

dt

dR i R i

dt

dV R i

dt

=

=

= +

(1)

0

0

0 0

d s m m d

f m m f

q s m q

L L L i

L L i

L L i

+ = +

(2)

The magnet is modeled by a current sourcemi parallel to

the resistancemR . This part of the circuit can be modeled as a

thevenin equivalent circuit, so that the direct axis equivalent

circuit of the machine will be the circuit shown in Fig. 4.

m mR i

mR0 q

mdLdd

dt

+

+

dV

difi

fd

dt

+

sR sl

Fig. 4. D axis thevenin equivalent circuit of the PMSM

This model is similar to the conventional equivalent circuitof the synchronous machine, except there is no leakage

inductance on the field.

C. Typical data for the PM motor

The data used for the simulations in this paper is from a 3-

phase 208-volt, 6 kW test motor used by the Sebastian and

Slemon [3]. Table 1 shows the value of the machine

parameters. The base values of the voltage and current are

RMS values of the stator phase voltage and current.

III. INTERFACE MODEL OF THE PMSM IN RTDS

Equations (1) and (2) can be used in the diagram of Fig.5to interface the machine to the network. In every time step the

program reads the voltage from the previous time step and

applies the Park transform to get the dq0 components of the

voltage, then using (1) it calculates the derivatives of the

fluxes. Predictor corrector integration calculates the new

values of the fluxes and by multiplying the inverse of

inductance matrix to the fluxes, dq0 component of the currents

can be obtained. Using the inverse Park transform we have the

phase currents ai , bi and ci that can be injected to the


3/6

network.

(from previous time step)

Eq (1)

di qi fi

Eq (2)

(for injection)

Park

Transform

Inverse

ParkTransform

, ,a b cv v v 0, ,d qv v v , ,d q f

, ,d q f , ,a b ci i i

Fig. 5. Interfacing the machine to the network.

IV. EMBEDDED MODEL OF THE PMSM IN RTDS

The machine can be modeled as set of mutual inductances

that change in value with time. In this case the model doesnt

have the problem of interface that may cause numerical

instabilities. This model doesnt use the Park transform and

directly solves the machine equations in phase domain. For a

machine, or in general, a set of time-varying mutual

inductances can be written:

( ) ( ) ( )( )d

v t L t i t dt

= (4)

Where v and i are vectors of node voltages and branch

currents and [ ]L is the inductance matrix of the set. Using the

trapezoidal integration we have:

[ ]

[ ] [ ] [ ]

1

1 1

( ) ( ) ( )2

( ) ( ) ( ) ( ) ( )2

Ih

ti t L t v t

tL t v t t L t L t t i t t

= +

+

(5)

The machine can be modeled as set of current sources Ih

parallel to a network of g values that can be obtained from the

matrix [ ]1

2

tGL L

= .

A. Calculating the Phase Domain Inductances of the

Permanent Magnet Machine

Equation (5) can be directly used to model the machine;

however we need to have the value of the inductances as a

function of time. Using an orthogonal transformation [5];

( ) ( )1 tT T = the data in table can be used to write:

( ) ( ) ( )1

0

0 0

0 0

0 0

d

abc q

L

L T L T

L

=

(6)

( )( )( )

( )1 00

af md

bf

cf

L L

L T

L

=

(7)

From (6) and (7) as an example we have:

( ) ( )cos 2 Haa s mL L L = + (8)

( ) ( ) ( )cos 2 H6

ab ba s mL L M L

= = +

(9)

( ) ( ) ( )cos Haf fa f L L M = = (10)

The rest of inductances can be calculated in a similar way.

The self inductance of the filed is the same in phase domain

and dq0 domain:

( )f fL L = (11)

Where:

( ) ( )0

0

1 1

3 3

1 2

3 2 3

s d q m d q

d q

s f md

L L L L L L L

L LM L M L

= + + =

+ = + =

(12)

Because the matrix has to be inverted in every time step,

its important to have a non-singular inductance matrix, so the

above parameters must satisfy this condition.

B. Inverting the Inductance Matrix of the Machine

The inductance matrix[ ]L of the machine has to be inverted

in every time step to calculate the G matrix of the machine. In

RTDS this task has to be done in a limited amount of time. As

a result having a fast routine for inverting the inductance

matrix is important. Cholesky decomposition can be used for

inductance matrices generated by FEM programs.

An analytical inverse of the inductance matrix can be

prepared when this matrix is as defined as the equations in the

preceding section.

V. COMPARING DIFFERENT MODELS OF THE PMSM IN RTDS

The dq0 and embedded model of the PM machine have

been implemented in RTDS in addition to the normal RTDS

synchronous machine model. Both of the machine models are

running at rated speed. It is possible to add a mechanical load

to the machine and project from the previous time step [6].

A. Performance of the Machines in Steady State

The circuit of the machine is shown in Fig. 6. A 60 Hz

three phase voltage source with 208 V line-line voltage

supplies the machines.The embedded model of the machine contains a component

with 4 coupled windings. The model allows entering the

parameters of the PM machine as input. The windings A, B

and C are Y connected armature windings with the neutral

isolated from the ground. A very large resistance provides the

isolation. The equivalent current source and resistance of the

magnet is modeled by a Thevenin equivalent circuit with the

1.272 kV voltage source in series with the magnet resistance.

The armature resistances are in series with the inductances.

Dq0 PM machine is connected to a similar AC source and


4/6

is a component with three nodes. Parameters of the machine

are identical, and the initial rotor angle ( )0 for both of them is

set to0 .

Embedded

Machine

Model

PM

Interfaced

MachineModel

AC source

AC source

m mR i

mR

sR

sR

sR

srsV

sV sr

Fig. 6. Test circuit for comparing the new machine models

There is an interfaced model of the synchronous machine

in RTDS. By setting the parameters, this machine is used as a

permanent magnet machine and the simulation results were

compared with the embedded and interfaced models of the

machine in Fig. 7.

Fig. 7 shows the steady state waveform of the phase A

voltage and current. The currents of the three machines are

exactly the same; although there is a 5S delay in the current

of dq0 machine model, which is the error of projection.

0 0 .005 0 .01 0 .015 0 .02 0 .025 0 .03 0 .035 0 .04 0 .045-200

-150

-100

-50

0

50

100

150

200

Time (Sec)

Voltage

(V)

-200

-150

-100

-50

0

50

100

150

200

Current(A)

Voltage

Currents

Fig. 7. Performance of the Machines in Steady State

The equivalent circuit of the machine in steady state is

shown in Fig. 8. The term 23

at the voltage source is

because of the Park transform that we used.

mV (

23m md m

V L i=

sRdXaIJG

aV (

Fig. 8. Equivalent circuit of the machine in steady state

Solving the circuit gives us the values of the armature

current at the steady state which agrees with the simulation

results.

,

,

23

79.953 (A)a peak md m

a peak

s

V L iI

jXd R

= =

+

( ((13)

B. Three Phase Short Circuit on the terminals of the Machine

Fig. 9. shows the current waveforms of the machines after

the short circuit. The currents of the machines are identical

however the delay of the current in dq0 model is more after

the short circuit.

0 0.05 0.1 0.15 0.2-100

-80

-60

-40

-20

0

20

40

60

80

100

Time (Sec)

Current(A)

Steady State Component

Transient Component

Fig. 9. Three phase short circuit current of the machines

Equation (14) gives the peak value of the short circuit

current at the steady state which agrees with the simulation

result.

( )2

2

, 2 2

240.721 (A)

3

mdsc peak m q s

d q s

LI i L R

L L R

= + =

+(14)

The transient part of the short circuit current contains three

components: steady state AC current, transient exponentiallydamping AC current with the time constant

dT , and the

transient exponentially damping DC component with the time

constanta

T, where:

0.084 (ms)s mddm

l LT

R = =

& (15)

1

1 1

2

a

s

d q

TR

L L

=

+

[7] (16)


5/6

We ignored the dampers in the machine so:

d q dL L L = = (17)

And

11.253 (ms)das

LT

R= = (18)

Equation (18) shows thatd

T the transient time constant of

the machine is very small, and its because mR the magnet

resistance is very large. This interesting fact removes the

transient AC component of the short circuit current, and we

just deal with the damping DC current and steady state

current. Fig. 9 verifies the time constants of the waveforms

obtained from the short circuit.

C. Numerical Stability of the Machines

Numerical stability of the different types of the permanent

magnet synchronous machine was studied by running the case

with different time steps. The embedded model of the machine

has the best performance among these models and is stable

even for very large time steps. This model is accurate and theresults are the same in different time steps. The RTDS

synchronous machine is unstable with the time-steps larger

than169 S , the same thing happens with the interface model

with the time-steps larger than167 S .Typical time-steps are

50 S . Similar results obtained when an inductive source

supplied the machines. The conventional model of the

machine becomes numerically unstable at large time-steps

with the inductive source. Poor performance of interface

machine in larger time steps is shown in Fig. 10.

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

40

Current(A)

Embedded model

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5x 10

5

Time(Sec)

Current(A)

Injection-based mod el

Fig. 10. Armature currents of the machines in time step 167 S

VI. FINITE ELEMENT APPROACH FORMODELING THE

ELECTRIC MACHINES IN RTDS

Equations (9)-(12) are approximate formula for calculating

the inductances of the synchronous machine however, in a real

machine inductances change as a more sophisticated function

of rotor position. Numerical methods like FEM base methods

can be used to model the electromagnetic phenomena in the

electric machines and also calculate the inductances, however

these programs are very slow and are not practical for network

solution.

In this paper the calculated inductances from [8] are

tabulated and used to run the embedded phase domain case of

the synchronous machine with more accurate inductances. The

machine is a 90kVA, 200V (L-L), 400 Hz, 4 pole, 3-phasesalient pole synchronous generator (aircraft generator). In

another case the traditional dq0 model of the synchronous

machine uses the simplified dq0 equivalent circuit of this

machine:

Fig. 12 shows the phase A open circuit voltage of both

machines which has a good agreement with the results in [9].

Fig. 12. Phase A open circuit voltages of the machines

Phase A armature current and the field current of the

machine after a 3 phase symmetric short circuit is shown in

Fig. 13. Fig. 13. (a) is the stator current of the dq0 machine

and Fig. 13. (b) is for the embedded phase domain machine

with real inductances.

0 5 10 15 20 25 30-500

0

500

1000

(a)

Time (ms)

Curr

ent(A)

0 5 10 15 20 25 30-500

0

500

1000

Time (ms)

Current(A)

(b)

0 5 10 15 20 25 30-50

0

50

100

Time (ms)

Current(A)

(c)

Fig. 13. The currents of the machine in short circuit a) phase current of dq0

model of the machine b) phase current of embedded model of the machine c)

Field current of embedded model of the machine

VII. CONCLUSION

The dq0 and embedded model of the permanent magnet

machine has been implemented and verified in RTDS. The

models have identical results. The embedded model of the

machine is more stable with large time-steps than the

conventional model of the machine. The problem of the

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5-200-100

0

100

200

Time (ms)Voltage(V)


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embedded model of the machine is the calculation load of the

model which is higher than the conventional model, because

the inductance matrix of the machine has to be inverted in

every time step. With the new RPC cards in RTDS, this task is

not a significant problem anymore. Coupling the FEM based

programs and transient simulation programs can be a very

successful step in modeling the power system network with

more accuracy.

VIII. APPENDIX

TABLEI

TYPICAL DATA FOR THE PMSM[SLEMON]

Parameter Symbol Typical value

Line-line rated voltage llV 208 V

Rated stator current lI 16.7 A

Rated power r a ted P 6016 W

D-axis inductance dL .249 pu 4.76 mH

Q-axis inductance qL .249 pu 4.76 mH

Stator leakage inductance sl .109 pu 2.09 mHStator resistance sR .059 pu 0.423

Magnet resistance mR 1.94 pu 13.92

Equivalent magnet current mi 5.47 pu 91.35 A

IX. ACKNOWLEDGEMENT

The authors gratefully acknowledge the support of RTDS

Technologies Inc.

X. REFERENCES

[1] L. Samaramayake and Y.K. Chin, "Speed synchronized control ofpermanent magnet synchronous motors with field-weakening,"

International Conference on Power and Energy Systems, EuroPES

2003, vol. 3, pp. 547-552, Sep. 2003.

[2] R. Kuffel, J. Giesbrecht, T. Maguire, R.P. Wierckx, and P.G. McLaren,

"A fully digital power system simulator operating in real time,"

Canadian Conference on Electrical and Computer Engineering, vol. 2,

pp. 733-736, 26-29 May 1996.

[3] T. Sebastian, and G.R. Slemon, "Transient modeling and performance of

variable-speed permanent-magnet motors, "IEEE Trans. Industry

Applications, Vol. 25, No.1, pp. 101-106, Sep. 1986.

[4] X. Longya, L. Ye, L. Zhen, and A. El-Antably, "A new design concept

of permanent magnet machine for flux weakening operation," IEEE

Trans. Industry Applications, vol. 31, No.2, pp. 373-378, Mar./Apr.

1995.

[5] P.M. Anderson and A.A. Fouad, Power system control and stability,

New York, IEEE Press, 1994, pp. 83-87.

[6] J.R. Marti and K.W. Louie, A phase-domain synchronous generator

model including saturation effects," IEEE Trans.Power Systems, vol.

12, No.2, pp. 222-229, Feb. 1997.

[7] B. Adkins and R.G. Harley, The general theory of alternating current

machines, London, Chapman and Hall, 1975, p. 162

[8] E. Deng, N.A.O. Demerdash, J.G. Vaidya, M.J. Shah, "A coupled finite-

element state-space approach for synchronous generators. II," IEEE

Trans. Aerospace and Electronic Systems, vol. 32, No.2, pp. 785 - 794,

Apr. 1996.

XI. BIOGRAPHIES

A. B. Dehkordi (M04) was born in Shahrekord,

Iran in 1977. He received his B.Sc. and M.Sc.

degrees in Electrical Engineering from Sharif

University of Technology, Tehran, Iran in 1999 and

2002 respectively. He worked as a research engineer

for Sharif University of Technology in power quality

projects in 2002. He is currently a Ph.D. student at

the Department of Electrical and Computer

Engineering at the University of Manitoba. Hisresearch is on the modeling of electric machines for

the Real Time Digital Simulation.

A. M. Gole (M82, SM04) obtained the B.Tech.

(EE) degree from IIT Bombay, India in 1978 and the

Ph.D. degree from the University of Manitoba,

Canada in 1982. He is currently a Professor of

Electrical and Computer Engineering at the

University of Manitoba. Dr. Goles research interests

include the utility applications of power electronics

and power systems transient simulation. As an

original member of the design team, he has made

important contributions to the PSCAD/EMTDC

simulation program. Dr. Gole is active on several working groups of CIGRE

and IEEE and is a Registered Professional Engineer in the Province of

Manitoba.

T. L. Maguire (M84, SM04)was born in Canadain 1951. He received a Ph.D. degree in 1992 at the

University of Manitoba. Dr Maguire is with the

RTDS technologies Inc. He is experienced with

FACTS and HVDC as well as with power system

control application and simulations.

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